Problem

Source: 2021 Israel TST Test 6 P1

Tags: sum of digits, TST, number theory



A pair of positive integers $(a,b)$ is called an average couple if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\]where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.